PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30EECS 42, Spring 2005 Week 3a 1AnnouncementsNew topics: Mesh (loop) method of circuit analysis Superposition method of circuit analysis Equivalent circuit idea (Thevenin, Norton) Maximum power transfer from a circuit to a load To stop blowing fuses in the lab, note how the breadboards are wired …EECS 42, Spring 2005 Week 3a 2Top view of boardEECS 42, Spring 2005 Week 3a 3Bottom view of board – note which way the wires goEECS 42, Spring 2005 Week 3a 4NODAL ANALYSIS(“Node-Voltage Method”)0) Choose a reference node 1) Define unknown node voltages2) Apply KCL to each unknown node, expressing current in terms of the node voltages=> N equations forN unknown node voltages3) Solve for node voltages=> determine branch currentsMESH ANALYSIS(“Mesh-Current Method”)1) Select M independent mesh currents such that at least one mesh current passes through each branch*M = #branches - #nodes + 12) Apply KVL to each mesh, expressing voltages in terms of mesh currents=> M equations forM unknown mesh currents3) Solve for mesh currents=> determine node voltagesPrimary Formal Circuit Analysis Methods*Simple method for planar circuitsA mesh current is not necessarily identified with a branch current.EECS 42, Spring 2005 Week 3a 51. Select M mesh currents.2. Apply KVL to each mesh.3. Solve for mesh currents.Mesh Analysis: Example #1EECS 42, Spring 2005 Week 3a 6Problem: We cannot write KVL for meshes a and b because there is no way to express the voltage drop across the current source in terms of the mesh currents.Solution: Define a “supermesh” – a mesh which avoids the branch containing the current source. Apply KVL for this supermesh. Mesh Analysis with a Current SourceiaibEECS 42, Spring 2005 Week 3a 7Eq’n 1: KVL for supermeshEq’n 2: Constraint due to current source:Mesh Analysis: Example #2iaibEECS 42, Spring 2005 Week 3a 8Mesh Analysis with Dependent Sources•Exactly analogous to Node Analysis•Dependent Voltage Source: (1) Formulate and write KVL mesh eqns. (2) Include and express dependency constraint in terms of mesh currents•Dependent Current Source: (1) Use supermesh. (2) Include and express dependency constraint in terms of mesh currentsEECS 42, Spring 2005 Week 3a 9Superposition Method (Linear Circuits Only)A linear circuit is constructed only of linear elements (linear resistors, linear dependent sources) and independent sources.Principle of Superposition:•In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone.Procedure:1. Determine contribution due to an independent source•Set all other sources to zero (voltage source short circuit; current source open circuit)2. Repeat for each independent source3. Sum individual contributions to obtain desired voltage or currentEECS 42, Spring 2005 Week 3a 10Superposition Example•Find Vo–+24 V2 4 4 A4 V+ –+Vo–EECS 42, Spring 2005 Week 3a 11EECS 42, Spring 2005 Week 3a 12Lecture 6, Slide 1EECS40, Spring 2004 Prof. SandersEquivalent Circuit Concept• A network of voltage sources, current sources, and resistors can be replaced by an equivalent circuit which has identical terminal properties (I-V characteristics) without affecting the operation of the rest of the circuit.+vA_network AofsourcesandresistorsiA?+vB_network BofsourcesandresistorsiBiA(vA) = iB(vB)EECS 42, Spring 2005 Week 3a 13•Voltage sources in series can be replaced by an equivalent voltage source:•Current sources in parallel can be replaced by an equivalent current source:Source Combinationsi1i2≡i1+i2–+–+v1v2≡–+v1+v2EECS 42, Spring 2005 Week 3a 14Thévenin Equivalent Circuit•Any* linear 2-terminal (1-port) network of indep. voltage sources, indep. current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent voltage source in series with a resistor without affecting the operation of the rest of the circuit.networkofsourcesandresistors≡–+VThRThRLiL+vL–abRLiL+vL–abThévenin equivalent circuit“load” resistorEECS 42, Spring 2005 Week 3a 15I-V Characteristic of Thévenin Equivalent•The I-V characteristic for the series combination of elements is obtained by adding their voltage drops:–+VThRThabii+vab–v = VTh+ iRI-V characteristic of resistor: v = iRI-V characteristic of voltage source: v = VThFor a given current i, the voltage drop vab is equal to the sum of the voltages dropped across the source (VTh)and the across the resistor (iRTh)vEECS 42, Spring 2005 Week 3a 16Finding VTh and RThOnly two points are needed to define a line. Choose two convenient points:1. Open circuit across terminals a,b i = 0, vab ≡ voc2. Short circuit across terminals a,b vab = 0, i ≡ -isc = -VTh/RTh–+VThRTh+voc = VTh––+VThRTh+voc = VTh–iv = VTh+ iRvab-iscvoc–+VThRThisciiEECS 42, Spring 2005 Week 3a 17Calculating a Thévenin Equivalent1. Calculate the open-circuit voltage, voc2. Calculate the short-circuit current, isc•Note that isc is in the direction of the open-circuit voltage drop across the terminals a,b !networkofsourcesandresistorsab+voc–networkofsourcesandresistorsabiscscocThocThivRvVEECS 42, Spring 2005 Week 3a 18Thévenin Equivalent ExampleFind the Thevenin equivalent with respect to the terminals a,b:EECS 42, Spring 2005 Week 3a 19EECS 42, Spring 2005 Week 3a 20Alternative Method of Calculating RThFor a network containing only independent sources and linear resistors:1. Set all independent sources to zerovoltage source short circuitcurrent source open circuit2. Find equivalent resistance Req between the terminals by inspectionOr, set all independent sources to zero1. Apply a test voltage source VTEST 2. Calculate ITESTeqRR ThTESTTESTThIVR network ofindependentsources andresistors, witheach sourceset to zeroReqnetwork ofindependentsources andresistors, witheach sourceset to zeroITEST–+VTESTEECS 42, Spring 2005 Week 3a 21RTh Calculation Example #1Set all independent sources to zero:EECS 42, Spring 2005 Week 3a 22Comments on Dependent SourcesA dependent source establishes a voltage or current whose value depends on the
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